Studying spin-$\frac{1}{2}$ fermions coupled to ${\mathbb{Z}}_{2}$ gauge fields on the square lattice, we show how a spatial modulation of the fermion hopping amplitude allows for the realization of various obstructed atomic insulators that host higher-order band topology. Including the effect of quantum dynamics of the gauge fields within a simplified model, we find a rich phase diagram of this system with a number of superfluid phases (which host distinct types of topological defects) arising from the attractive interactions meditated by the gauge fields. The evolution of the superfluid obtained by the destabilization of the obstructed atomic insulators from the Bardeen-Cooper-Schrieffer (BCS) type to a Bose-Einstein condensate (BEC) of tightly bound pairs occurs via the realization of these different superfluid phases separated by first-order transitions.
Studying spin-21 fermions coupled to Z2 gauge fields on the square lattice, we show how a spatial modulation of the fermion hopping amplitude allows for the realization of various obstructed atomic insulators that host higher-order band topology. Including the effect of quantum dynamics of the gauge fields within a simplified model, we find a rich phase diagram of this system with a number of superfluid phases (which host distinct types of topological defects) arising from the attractive interactions meditated by the gauge fields. The evolution of the superfluid obtained by the destabilization of the obstructed atomic insulators from the Bardeen-Cooper-Schrieffer (BCS) type to a Bose-Einstein condensate (BEC) of tightly bound pairs occurs via the realization of these different superfluid phases separated by first-order transitions.
We study two models realized by two-component Fermi gases loaded in optical lattices. We clarify that multiband effects inevitably caused by the optical lattices generate a rich structure, when the systems crossover from the region of weakly bound molecular bosons to the region of strongly bound atomic bosons. Here the crossover can be controlled by attractive fermion interaction. One of the present models is a case with attractive fermion interaction, where an insulator-superfluid transition takes place. The transition is characterized as the transition between a band insulator and a Bose-Einstein condensate superfluid state. Differing from the conventional Bardeen-Cooper-Schrieffer (BCS) superfluid transition, this transition shows unconventional properties. In contrast to the one-particle excitation gap scaled by the superfluid order parameter in the conventional BCS transition, because of the multiband effects, a large gap of one-particle density of states is retained all through the transition, although the superfluid order grows continuously from zero. A re-entrant transition with lowering temperature is another unconventionality. The other model is the case with coexisting attractive and repulsive interactions. Within a mean-field treatment, we find a new insulating state, an orbital ordered insulator. This insulator is one candidate for the Mott insulator of molecular bosons and is the first example that the orbital internal degrees of freedom of molecular bosons appears explicitly. Besides the emergence of a new phase, a coexisting phase also appears where superfluidity and an orbital order coexist just by doping holes or particles. The insulating and superfluid particles show differentiation in momentum space as in the high-Tc cuprate superconductors.
We theoretically investigate a tight-binding model of fermions hopping on the square-octagon lattice which consists of a square lattice with plaquette corners themselves decorated by squares. Upon the inclusion of second-neighbor spin-orbit coupling or non-Abelian gauge fields, time-reversal symmetric topological Z2 band insulators are realized. Additional insulating and gapless phases are also realized via the non-Abelian gauge fields. Some of the phase transitions involve topological changes to the Fermi surface. The stability of the topological phases to various symmetry-breaking terms is investigated via the entanglement spectrum. Our results enlarge the number of known exactly solvable models of Z2 band insulators and are potentially relevant to the realization and identification of topological phases in both the solid-state and cold atomic gases.
We consider a two-component Fermi gas with attractive interactions on a square optical lattice, and study the interplay of Zeeman field, spin-orbit coupling, and next-nearest-neighbor hopping on the ground-state phase diagrams in the entire BCS-BEC evolution. In particular, we first classify and distinguish all possible superfluid phases by the momentum-space topology of their zero-energy quasiparticle-quasihole excitations, and then numerically establish a plethora of quantum phase transitions in between. These transitions are further signaled and evidenced by the changes in the corresponding topological invariant of the system, i.e., its Chern number. Lastly, we find that the superfluid phase exhibits a reentrant structure, separated by a fingering normal phase, the origin of which is traced back to the changes in the single-particle density of states.
Since the realization of Bose-Einstein condensation in 1995, experimentalists in ultracold atomic gas community have developed a set of entirely new methods to study quantum gases. In ultracold atomic gases, implementation of synthetic gauge fields, the spin-orbit coupling, optical lattices and Feshbach resonances, provides a clean and highly controllable platform to tune the parameters of the system, which are typically fixed for real materials. Motivated by the rapid developments of experimental progresses, I investigate the ultracold atomic gases with synthetic gauge fields. \n \nIn this thesis, a bosonic gas in optical lattices in the presence of an effective staggered magnetic flux will be studied, using a generalized Bose-Hubbard model. Phase transition from the Mott insulating phase to the superfluid phase will be discussed. Novel superfluid phases including the stripe phase and the plane wave phase will be discussed. Furthermore, the existence of a tricritical point in the phase diagram is predicted. \n \nTo study Fermi systems with synthetic gauge fields, a spin-1=2 Fermi gas with Rashba spin-orbit coupling is investigated by studying the structure of vortex line through the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation (BCSBEC) crossover. Due to the spin-orbit coupling, the s-wave interacting Fermi gas behaves like an effective p-wave paired system. A discussion about the vortex line structure will be provided by calculating the spatial distribution of the gap function, the density, the particle current, and also the elementary excitations. It has been found that the most robust superfluid occurs around the unitary point, indicated by the smallest healing length and the largest critical current. \n \nFinally, motivated by the recent progress of p-wave resonantly Fermi gas in experiment, an extension of Nozières-Schmitt-Rink method to the p-wave case will be given. An investigation of the normal state properties will be given by calculating the equation of state and the p-wave contacts, which have been measured by the experiment. Furthermore, the superfluid transition temperature with realistic parameters appropriate for experiment is calculated and is found to be within experimental reach.
Motivated by the recent realization of the Qi-Wu-Zhang (QWZ) model for the minimal spin-orbit-coupled quantum anomalous Hall phase in ultracold fermions, we introduce the on-site attractive interaction and map out the topological phase diagram for the Qi-Wu-Zhang-Hubbard (QWZH) model at the mean-field level. We show that, in addition to the conventional chiral topological superfluids (TSs) with nonzero Chern number, there is also an obstructed superfluid (OSF) with zero Chern number in the extended region of phase space. This phase is characterized by gapless edge states completely detaching from the bulk energy bands and two Z2 numbers. We find the gapless edge states in OSF are protected by particle-hole symmetry and robust for the disorder. The appearance of OSF demonstrates that it is not sufficient to label TSs in the two-dimensional D class only by Chern numbers. These different topological phases in the attractive QWZH model can be detected experimentally by spin-resolved time-of-flight imaging. We also plot the phase diagram of the attractive QWZH model with one-dimensional spin-orbit coupling and find a series of TSs with gapless Dirac points. Considering the successful realization of the QWZ model and the tremendous progress in manipulating cold atom systems, our work provides a possible route to realize OSF in cold fermionic systems in the future.
This thesis concerns ultracold two-dimensional atomic gases and the associated novel superfluid phases in an optical lattice subjected to staggered rotations. In the first part, we study the Berezinskii-Kosterlitz-Thouless (BKT) transition in a two-dimensional Bose gas. Motivated by recent experiments on the cold gas in the BKT regime, we employ a modified Popov theory and a renormalization group theory to describe the system. The two-dimensional renormalization group flow equations display unique features, which we interpret as precursors of the BKT physics. We further calculate higher correlation functions close to the BKT transition, which can serve as observables for detecting effects beyond mean-field. Finally, we obtain the density profile under the conditions of the experiment performed in the group of Dalibard in Paris, France. In the second part of the thesis, we study a recent experimental proposal of a time-dependent optical lattice with staggered rotations. We show that in the tight-binding regime, the system can be effectively described by a (Bose-)Hubbard model subjected to an artificial staggered magnetic field. The staggered magnetic field (flux) in the lattice leads to conical points in the band structure which possesses distinct minima for different strength, and can give rise to Dirac-like dispersions. For bosons, we show that besides the homogenous superfluid phase, a staggered-vortex superfluid phase is stabilized for large flux values. We show that the staggered flux renormalizes the phase boundary and find a multi-critical point separating the two superfluid phases and the Mott insulating phase. Finally, the full phase diagram of the generalized Bose-Hubbard model at zero temperature is determined. We then generalize the model for a mixture of ultracold bosons and spin-1/2 fermions. By using the Bogoliubov theory for the bosonic sector, we show that the bosonic superfluid mediates attractive interactions between fermions up to the nearest-neighbor site. The resulting model is an extended Hubbard model in the presence of a staggered magnetic field, where the on-site and the nearest-neighbor attractive interactions can be tuned independently. We then employ the Bardeen-Cooper-Schrieffer theory for superconductivity, by considering an on-site pairing order parameter competing with a nearest-neighbor resonating-valence-bond order parameter. Due to conical points in the energy band, the system possesses quantum critical property and the resulting phase diagram displays features which are strikingly similar to that of high-critical-temperature superconductors or heavy-fermion superconductors.